let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)

for a, u being Function of Y,BOOLEAN

for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let G be Subset of (PARTITIONS Y); :: thesis: for a, u being Function of Y,BOOLEAN

for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let a, u be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let PA be a_partition of Y; :: thesis: All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a '&' u),PA,G)) . z = TRUE or ((Ex (a,PA,G)) '&' u) . z = TRUE )

A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

assume A2: (All ((a '&' u),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) '&' u) . z = TRUE

A3: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

( a . x = TRUE & u . x = TRUE )

for a, u being Function of Y,BOOLEAN

for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let G be Subset of (PARTITIONS Y); :: thesis: for a, u being Function of Y,BOOLEAN

for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let a, u be Function of Y,BOOLEAN; :: thesis: for PA being a_partition of Y holds All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let PA be a_partition of Y; :: thesis: All ((a '&' u),PA,G) '<' (Ex (a,PA,G)) '&' u

let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not (All ((a '&' u),PA,G)) . z = TRUE or ((Ex (a,PA,G)) '&' u) . z = TRUE )

A1: z in EqClass (z,(CompF (PA,G))) by EQREL_1:def 6;

assume A2: (All ((a '&' u),PA,G)) . z = TRUE ; :: thesis: ((Ex (a,PA,G)) '&' u) . z = TRUE

A3: for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

( a . x = TRUE & u . x = TRUE )

proof

A4:
((Ex (a,PA,G)) '&' u) . z = ((Ex (a,PA,G)) . z) '&' (u . z)
by MARGREL1:def 20;
let x be Element of Y; :: thesis: ( x in EqClass (z,(CompF (PA,G))) implies ( a . x = TRUE & u . x = TRUE ) )

assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: ( a . x = TRUE & u . x = TRUE )

then (a '&' u) . x = TRUE by A2, BVFUNC_1:def 16;

then (a . x) '&' (u . x) = TRUE by MARGREL1:def 20;

hence ( a . x = TRUE & u . x = TRUE ) by MARGREL1:12; :: thesis: verum

end;assume x in EqClass (z,(CompF (PA,G))) ; :: thesis: ( a . x = TRUE & u . x = TRUE )

then (a '&' u) . x = TRUE by A2, BVFUNC_1:def 16;

then (a . x) '&' (u . x) = TRUE by MARGREL1:def 20;

hence ( a . x = TRUE & u . x = TRUE ) by MARGREL1:12; :: thesis: verum

per cases
( for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

u . x = TRUE or ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) ) ;

end;

u . x = TRUE or ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & not u . x = TRUE ) ) ;

suppose A5:
for x being Element of Y st x in EqClass (z,(CompF (PA,G))) holds

u . x = TRUE ; :: thesis: ((Ex (a,PA,G)) '&' u) . z = TRUE

end;

u . x = TRUE ; :: thesis: ((Ex (a,PA,G)) '&' u) . z = TRUE

now :: thesis: ((Ex (a,PA,G)) '&' u) . z = TRUE end;

hence
((Ex (a,PA,G)) '&' u) . z = TRUE
; :: thesis: verumper cases
( ex x being Element of Y st

( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) or for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ;

( x in EqClass (z,(CompF (PA,G))) & a . x = TRUE ) or for x being Element of Y holds

( not x in EqClass (z,(CompF (PA,G))) or not a . x = TRUE ) ) ;

end;